''I'll wait for you,'' historian and lecturer Henryk calls as the ship's science centre empties faster than a safety drill, and 500 passengers head for the decks. It's the second polar bear sighting of the day, and this time it's a prized view of the animal feasting on a seal.

I'm cruising on the Roald Amundsen to visit the remote and wild east coast of Greenland, but it's quickly becoming clear that in the Arctic, the best things often happen at sea. My journey on Hurtigruten's 12-day ''Greenland and Iceland: the Ultimate Fjord and National Park Experience'' begins in Longyearbyen, the world's northernmost town, on the Norwegian archipelago of Svalbard. It's early morning, which feels irrelevant in an Arctic summer, when the sun never sets.

As we sail from Longyearbyen through the large fiord that almost slices Svalbard in two, little auks fly ahead of the ship, seemingly bouncing across the water, the birds' bellies too full for them to get airborne. Near the mouth of the fiord, other seabirds spiral like a twister in a feeding frenzy.

Within two hours of leaving Longyearbyen, there's only horizon and open sea in front of the ship. By mid-afternoon, with Svalbard still large behind us, I'm standing at the bow of the ship when there are sprays of water ahead and the backs of two fin whales, the second-largest animals on earth, emerge from the sea.

For the two days across the Greenland Sea to the frayed east coast of the country, whales regularly appear. There are telltale distant splashes of breaching whales, and when I sit up in bed in the middle of one bright night, there's the spray of a surfacing whale immediately outside my cabin.

The underlying appeal of the Arctic is that it's a world so foreign as to be almost alien. Polar bears outnumber trees, and icebergs float about the sea like pleasure craft. On the morning we approach the Greenland coast, I wake to the sight of the ship surrounded by hundreds of ice floes - escapees from the pack ice that grips the northern part of the country. The bare mountains of Greenland rise in the distance, seeming to hover ethereally above the sea mist.

The polar silence is broken only by the sound of floes rattling against Roald Amundsen's reinforced hull as it pushes through this icy obstacle course. The ship has slowed to little more than one knot, and we creep towards Greenland at a suitably glacial speed.

The sky is clear, even of contrails - we haven't seen a plane since leaving Svalbard - and an app that charts the position of every ship in the world shows we're the only vessel along the entire east coast. It might be lonely if it wasn't so wonderful.

The ice floes bring their own treasures. A ring seal watches us pass from the edge of one floe, and a polar bear strolls across another, diving into the sea and swimming between floes as the ship approaches. It's our first sighting of a polar bear, and if 500 passengers hurrying to the bow could ever overturn a 21,000-tonne ship, this would be the moment. A few hours later there's a second bear - the sighting that trumped a history lecture. It won't be the last time polar bears disrupt the ship's schedule.

The next morning, with Roald Amundsen nudged in against the coast of this brown island known optimistically as Greenland, another polar bear is sighted on a narrow, rocky island ahead of the ship. There are said to be just 1500 polar bears along Greenland's east coast, and in 24 hours we've seen three of them. As the bear swims away, it's not the only thing that disappears. With it go our plans of a kayaking excursion from the ship, for the simple reason that, to a bear, we might resemble a floating food market. But what one Arctic creature takes away, another brings. Less than an hour later, a pod of narwhals - the Arctic whales that resemble unicorns - hunts along the shores, splashing and herding, their tusks rising from the water like broomsticks.

''Best day ever,'' I hear one passenger declare, and it's still only 8am.

The sea ice increases and gets only more complex as we cruise south towards Ittoqqortoormiit, one of only two towns on the east coast and Greenland's most isolated settlement.

At times the ice is like the lane ropes of a swimming pool as the captain finds long channels between floes, but at other times it stretches all the way to the horizon. It's a chaotic sight, all this ice broken and scattered into pieces, like a jigsaw puzzle fresh out of its box.

Soon we're paddling along the shores of Hekla Havn, a natural harbour cut into the shores of Danmark Island. Across the bay, a five-kilometre-wide glacier curls through the mountains, and dozens of icebergs bob about the harbour, creating a kind of natural slalom course for our kayaks. The morning is so calm that we paddle across our own reflections, keeping a distance from the shore because of the ever-present possibility of polar bears.

As we kayak laps around a couple of large icebergs, Arctic terns spear into the ocean around us, hunting seafood snacks. A few hundred metres away there's a thunderous crack as ice calves away from the largest iceberg in the harbour.

From Scoresby Sund there remains one more Arctic sea crossing as the ship sets off for Iceland, a day's sailing almost due south of our current position. Night creeps in for the first time, with the sun finally setting, or at least momentarily resting below the horizon. The sea is finally clear of ice, but so much else remains in these waters.

Over dinner in Aune restaurant, as we near the Arctic Circle, a couple of passengers rise from their chairs to peer out the tall windows of the restaurant. Orcas have been sighted. I scan the seas but can't spot the animals, yet I know they're out there. This is the Arctic, after all.